#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Time    : 2020/6/3 13:03
# @USER    : Shengji He
# @File    : New21Game.py
# @Software: PyCharm
# @Version  : Python-
# @TASK:


class Solution:
    def new21Game(self, N: int, K: int, W: int) -> float:
        """
        Alice plays the following game, loosely based on the card game "21".

        Alice starts with 0 points, and draws numbers while she has less than K points.  During each draw,
        she gains an integer number of points randomly from the range [1, W], where W is an integer.
        Each draw is independent and the outcomes have equal probabilities.

        Alice stops drawing numbers when she gets K or more points.
        What is the probability that she has N or less points?

        :param N: int
        :param K: int
        :param W: int
        :return: float

        Example 1:
            Input: N = 10, K = 1, W = 10

            Output: 1.00000

            Explanation:  Alice gets a single card, then stops.

        Example 2:
            Input: N = 6, K = 1, W = 10

            Output: 0.60000

            Explanation:  Alice gets a single card, then stops.
            In 6 out of W = 10 possibilities, she is at or below N = 6 points.

        Example 3:
            Input: N = 21, K = 17, W = 10

            Output: 0.73278

        Note:
            - 0 <= K <= N <= 10000
            - 1 <= W <= 10000
            - Answers will be accepted as correct if they are within 10^-5 of the correct answer.
            - The judging time limit has been reduced for this question.
        """
        if K == 0:
            return 1.0
        dp = [0.] * (K + W)
        dp[K: 1 + min([N, K - 1 + W])] = (1.0,) * (1 + min([N, K - 1 + W]) - K)
        dp[K - 1] = min([N - K + 1, W]) / W
        for x in range(K - 2, -1, -1):
            dp[x] = dp[x + 1] - (dp[x + W + 1] - dp[x + 1]) / W
        return round(dp[0], 5)


if __name__ == '__main__':
    S = Solution()
    N, K, W = 6, 1, 10
    print(S.new21Game(N, K, W))
    print('done')
